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Four SEI endemic models with periodicity and separatrices

L Q Gao1, J Mena-Lorca, H W Hethcote

  • 1Department of Mathematics, North Central College, Naperville, Illinois, USA.

Mathematical Biosciences
|July 1, 1995
PubMed
Summary

Certain infectious disease models (SEI) with simple mass action incidence can exhibit periodic solutions. However, models with standard mass action incidence do not show periodic solutions, indicating key differences in disease dynamics.

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Area of Science:

  • Epidemiology
  • Mathematical Biology
  • Infectious Disease Modeling

Background:

  • Periodic solutions are known for some Susceptible-Infectious-Exposed (SI and SEI) models.
  • Understanding model features that lead to periodic solutions is crucial for predicting disease dynamics.

Purpose of the Study:

  • To investigate four SEI models with varying reproduction rates (disease-reduced or uniform).
  • To identify specific model characteristics that promote or prevent periodic solutions in SEI models.

Main Methods:

  • Analysis of four SEI models, differentiating between simple mass action (beta XY) and standard mass action (lambda XY/N) incidence.
  • Examination of models with disease-reduced and uniform reproduction rates.
  • Investigating equilibrium stability and manifold behavior in specific SEI models.

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Main Results:

  • SEI models employing simple mass action incidence (beta XY) can yield periodic solutions under certain parameter values.
  • SEI models utilizing standard mass action incidence (lambda XY/N) do not exhibit periodic solutions.
  • In a specific SEI model (lambda XY/N, uniform reproduction), intermediate lambda values create a saddle point, separating disease-free and susceptible-free equilibria attractors.

Conclusions:

  • The form of incidence (simple vs. standard mass action) significantly influences the potential for periodic solutions in SEI models.
  • Standard mass action incidence in SEI models appears to preclude periodic solutions.
  • Complex dynamics, including stable manifold separation, can occur in SEI models even without periodic solutions.